1. The problem statement, all variables and given/known data (Note: This isn't an assignment problem, more a curiosity about the derivation of an equation - hopefully it is still posted in the right forum..) I have working through the derivation for the partial molar Gibbs free energy of mixing from the Flory-Huggins expression for the Gibbs free energy of mixing, however my math skills are (very) rusty - especially when it comes to partial derivatives. 2. Relevant equations Gibbs Free Energy of Mixing (Flory-Huggins Theory): ΔGM=R·T[n1·ln(ϕ1)+n2·ln(ϕ2)+n1· ·ϕ2] partial molar Gibbs free energy of dilution: Δμ1=R·T[ln(1-ϕ2)+(1-1/r)·ϕ2 + ·ϕ22] where KB=Boltzmann Constant, ni=number of moles of i, ϕi=volume fraction of i, =Flory-Huggins Interaction Parameter ϕ1=N1/N0=N1/(N1+r·N2) ϕ2=(r·N2)/N0=(r·N2)/(N1+r·N2) N0=N1+r·N2 where N0 is the number of lattice sites N1 is the number of solvent molecules N2 is the number of polymer molecules, each occupying "r" lattice sites (or "r" segments) and R=KB·NA where NA= Avogadro's constant 3. The attempt at a solution ΔGM=R·T[n1·ln(ϕ1)+n2·ln(ϕ2)+n1· ·ϕ2] applying R=KB·NA ΔGM=KB·T[N1·ln(ϕ1)+N2·ln(ϕ2)+N1· ·ϕ2] expressing ϕ1 and ϕ2 in terms of N1 and N2 gives ΔGM=KB·T[N1·ln(N1/(N1+r·N2))+N2·ln((r·N2)/(N1+r·N2))+N1· ·(r·N2)/(N1+r·N2)] Next step would be to take the partial derivative of the equation with respect to N1, I've tried many times but I cannot get anything close to the equation for the partial molar Gibbs free energy of dilution.. I broke it up, letting Ψ1=N1·ln(N1/(N1+r·N2)) Ψ2=N2·ln((r·N2)/(N1+r·N2)) Ψ3=N1· ·(r·N2)/(N1+r·N2) taking the partial derivative dΨ1/dN1=ln(N1/(N1+rN2)+[1-(N1+rN1N2)/(N1+rN2)] dΨ2/dN1=-N2[(1+rN2)/(N1+rN2)] dΨ2/dN1= [(rN2)/(N1+rN2)]+N1 [(-rN2)/(N1+rN2)2 which when I plug it all back in, gives me a big mess..